Nonlinear compressible vortex sheets in two space dimensions

Nonlinear compressible vortex sheets in two space dimensions Jean-Franc¸ois Coulombel†, and Paolo Secchi‡ January 8, 2007 † Team SIMPAF of INRIA Futurs, CNRS and Universite Lille 1, Laboratoire Paul Painleve, Cite Scientifique 59655 VILLENEUVE D'ASCQ CEDEX, France ‡ Dipartimento di Matematica, Facolta di Ingegneria, Via Valotti, 9, 25133 BRESCIA, Italy Emails: , Abstract We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nev- ertheless, we prove the local in time existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized equations exhibit a loss of regularity, our existence result is proved by using a suitable modification of the Nash-Moser iteration scheme. We also show how a similar analysis yields the existence of weakly sta- ble shock waves in isentropic gas dynamics, and the existence of weakly stable liquid/vapor phase transitions. AMS subject classification: 76N10, 35Q35, 35L50, 76E17. Keywords: compressible Euler equations, vortex sheets, contact discontinuities, weak stability, loss of derivatives, Nash-Moser iteration scheme.

• nonlinear problem


• nash-moser iteration


• shock waves


• weakly stable


• planar compressible


• vortex sheets


• called uniform stability


• euler equations